#### Volume 23, issue 7 (2019)

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The fundamental group of compact Kähler threefolds

### Benoît Claudon, Andreas Höring and Hsueh-Yung Lin

Geometry & Topology 23 (2019) 3233–3271
##### Abstract

Let $X$ be a compact Kähler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that ${\pi }_{1}\left(X\right)\simeq {\pi }_{1}\left(Y\right)$. We also prove the bimeromorphic existence of algebraic approximations for compact Kähler manifolds of algebraic dimension $dimX-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact Kähler threefolds.

##### Keywords
fundamental group, compact Kähler manifolds, algebraic approximations, elliptic fibrations
##### Mathematical Subject Classification 2010
Primary: 14D07, 32J17, 32J27, 32Q55
##### Publication
Received: 25 July 2017
Revised: 20 August 2018
Accepted: 29 April 2019
Published: 30 December 2019
Proposed: Dan Abramovich
Seconded: Anna Wienhard, John Lott
##### Authors
 Benoît Claudon Université Rennes 1 IRMAR-UMR 6625 Rennes France Institut Universitaire de France Andreas Höring Université Côte d’Azur CNRS Laboratoire Jean-Alexandre Dieudonné Nice France Institut Universitaire de France Hsueh-Yung Lin Mathematisches Institut der Universität Bonn Bonn Germany